Rigidity, Volume, and Combinatorics in Hyperbolic Geometry

双曲几何中的刚度、体积和组合学

基本信息

  • 批准号:
    1608759
  • 负责人:
  • 金额:
    $ 24万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2016
  • 资助国家:
    美国
  • 起止时间:
    2016-09-01 至 2018-09-30
  • 项目状态:
    已结题

项目摘要

For much of the twentieth century, the challenge to describe the shape of three dimensional spaces was viewed as an algebraic problem: indeed it is through algebra that we first distinguish the (topological) structure of a sphere from that of a doughnut. Work of William Thurston brought the geometry of such spaces more clearly into view as a central feature to explore. The triumph of this approach was Perelman's proof of Thurston's geometrization conjecture, that each three-dimensional manifold could be naturally broken up into pieces, each with a uniform geometry. The study of these geometries is frequently reduced, via a notion of rigidity, to considering simple combinatorial structures arising from loops on surfaces. The proposed research will explore how these structures predict volume, diameter, length and other aspects of the geometry, relating to notions from quantum physics.A longstanding connection between volume of hyperbolic three-manifolds and Weil-Petersson distance relied on a combinatorial comparison via the pants graph which organizes maximal multicurves on a surface, yet other connections were known to arise from work of Witten via renormalized volume. Recent work of Schlenker made this connection explicit in the context of quasi-Fuchsian manifolds, and PI's proposed work will develop this idea further to explicitly relate fibered 3-manifolds and translation distiance. More generally, a primary project in the proposed research is to solidify and extend the connection between combinatorics and geometry in closed manifolds, and to continue to investigate the structure of deformation spaces of Kleinian groups with these tools. As an example of the power of these techniques, bi-Lipschitz models for 'random' Heegaard splittings provide a full solution (with Rivin and Souto) to the conjecture of Dunfield and Thurston that random Heegaard splittings are almost surely hyperbolic and their volume grows linearly. Finally, the PI will pursue projects with Minsky, and with Modami, Leininger and Rafi on the role of combinatorics in the geometry of geodesics in the Weil-Petersson metric, investigating disparities between Teichmuller and Weil-Petersson geodesics in terms of unique ergodicity.
在世纪的大部分时间里,描述三维空间形状的挑战被视为一个代数问题:事实上,正是通过代数,我们第一次区分了球体和甜甜圈的(拓扑)结构。威廉·瑟斯顿(William Thurston)的作品使这种空间的几何形状更清晰地成为探索的中心特征。这种方法的胜利是佩雷尔曼对瑟斯顿几何化猜想的证明,即每个三维流形都可以自然地分解成碎片,每个碎片都有统一的几何形状。对这些几何的研究经常通过刚性的概念被简化为考虑由曲面上的环产生的简单组合结构。拟议中的研究将探索这些结构如何预测体积,直径,长度和几何的其他方面,与量子物理学的概念有关。双曲三流形的体积和Weil-Petersson距离之间的长期联系依赖于通过裤子图的组合比较,该图组织表面上的最大多曲线,但其他联系已知来自维滕通过重整化体积的工作。Schlenker最近的工作在准Fuchsian流形的上下文中明确了这种联系,PI的提议工作将进一步发展这种想法,以明确地将纤维3-流形和平移distiance联系起来。更一般地说,拟议研究的主要项目是巩固和扩展闭流形中组合学和几何学之间的联系,并继续研究Kleinian群变形空间的结构。作为这些技术的力量的一个例子,双Lipschitz模型为“随机”Heegaard分裂提供了一个完整的解决方案(与Rivin和Souto)的猜想邓菲尔德和瑟斯顿,随机Heegaard分裂几乎肯定是双曲线和他们的体积线性增长。最后,PI将与Minsky以及Modami,Leininger和Rafi一起开展项目,研究组合数学在Weil-Petersson度量中的测地线几何中的作用,调查Teichmuller和Weil-Petersson测地线在独特遍历性方面的差异。

项目成果

期刊论文数量(0)
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会议论文数量(0)
专利数量(0)

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Jeffrey Brock其他文献

Tameness on the boundary and Ahlfors’ measure conjecture
  • DOI:
    10.1007/s10240-003-0018-y
  • 发表时间:
    2003-12-01
  • 期刊:
  • 影响因子:
    3.500
  • 作者:
    Jeffrey Brock;Kenneth Bromberg;Richard Evans;Juan Souto
  • 通讯作者:
    Juan Souto

Jeffrey Brock的其他文献

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{{ truncateString('Jeffrey Brock', 18)}}的其他基金

REU Site: Summer Undergraduate Math Research at Yale
REU 网站:耶鲁大学暑期本科生数学研究
  • 批准号:
    2050398
  • 财政年份:
    2021
  • 资助金额:
    $ 24万
  • 项目类别:
    Standard Grant
Rigidity, Volume, and Combinatorics in Hyperbolic Geometry
双曲几何中的刚度、体积和组合学
  • 批准号:
    1849892
  • 财政年份:
    2018
  • 资助金额:
    $ 24万
  • 项目类别:
    Standard Grant
Mapping Class Groups and Teichmuller Theory, May 7-14, 2014
映射类组和 Teichmuller 理论,2014 年 5 月 7-14 日
  • 批准号:
    1439369
  • 财政年份:
    2014
  • 资助金额:
    $ 24万
  • 项目类别:
    Standard Grant
Combinatorics, Models, and Bounds in Hyperbolic Geometry
双曲几何中的组合学、模型和界限
  • 批准号:
    1207572
  • 财政年份:
    2012
  • 资助金额:
    $ 24万
  • 项目类别:
    Continuing Grant
Teichmüller Theory, Kleinian Groups, and the Complex of Curves
泰希米勒理论、克莱尼群和曲线复形
  • 批准号:
    0906229
  • 财政年份:
    2009
  • 资助金额:
    $ 24万
  • 项目类别:
    Standard Grant
Focused Research Group: Collaborative Research: Geometry and Deformation Theory of Hyperbolic 3-Manifolds
重点研究组:合作研究:双曲3流形的几何与变形理论
  • 批准号:
    0553694
  • 财政年份:
    2006
  • 资助金额:
    $ 24万
  • 项目类别:
    Standard Grant
Effective Rigidity, Combinatorial Models, and Parameter Spaces for Low-Dimensional Hyperbolic Manifolds
低维双曲流形的有效刚性、组合模型和参数空间
  • 批准号:
    0505442
  • 财政年份:
    2005
  • 资助金额:
    $ 24万
  • 项目类别:
    Standard Grant
The Classification Problem for Hyperbolic 3-Manifolds
双曲 3 流形的分类问题
  • 批准号:
    0354288
  • 财政年份:
    2003
  • 资助金额:
    $ 24万
  • 项目类别:
    Continuing Grant
The Classification Problem for Hyperbolic 3-Manifolds
双曲 3 流形的分类问题
  • 批准号:
    0204454
  • 财政年份:
    2002
  • 资助金额:
    $ 24万
  • 项目类别:
    Continuing Grant
Mathematical Sciences Postdoctoral Research Fellowships
数学科学博士后研究奖学金
  • 批准号:
    9706007
  • 财政年份:
    1997
  • 资助金额:
    $ 24万
  • 项目类别:
    Fellowship Award

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